On the parametrization of self-similar and other fractal sets

We prove for many self-similar, and some more general, sets $E \subset \mathbb{R}^{n}$ that if $s$ is the Hausdorff dimension of $E$ and $f: \mathbb{R}^{m} \to \mathbb{R}^{n}$ is Hölder continuous with exponent $m/s$, then the $s$-dimensional Hausdorff measure of $E \cap f(\mathbb{R}^{m})$ is $0$.